On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic

TL;DR

This paper investigates the decidability of certain restricted expansions of Presburger arithmetic with polynomial predicates, showing positive results for single-variable cases involving perfect powers and low-degree polynomials.

Contribution

It provides new decidability results for single-variable theories with specific polynomial predicates, extending understanding of Presburger arithmetic expansions.

Findings

Decidability for perfect fixed powers via hyperelliptic Diophantine equations.

Decidability for polynomials of degree at most three using low genus algebraic curves.

Limitations and hardness results when restrictions are lifted.

Abstract

We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form $2n^3-5n+3$, etc.) Although the full attendant first-order theories are well known to be undecidable, very little is known when one restricts the number of variables. In the case of single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability ofthe corresponding theory follows from the solvability of hyperellipticDiophantine equations; and (ii) for polynomials of degree at most three, we establish decidability by relying on the low genus of the resulting algebraic curves. Finally, we discuss limitations and hardness results (via encodings of longstanding open Diophantine problems) as soon as any of the above restrictions are lifted.